Measure Theory 1-5

3247 Pages·23.851 MB·English

by  D. H. Fremlin

MEASURE THEORY
Volume 1
D.H.Fremlin
By the same author:
Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.
Consequences of Martin’s Axiom, Cambridge University Press, 1982.
Companions to the present volume:
Measure Theory, vol. 2, Torres Fremlin, 2001;
Measure Theory, vol. 3, Torres Fremlin, 2002;
Measure Theory, vol. 4, Torres Fremlin, 2003;
Measure Theory, vol. 5, Torres Fremlin, 2008.
First edition May 2000
Second edition January 2011
MEASURE THEORY
Volume 1
The Irreducible Minimum
D.H.Fremlin
Research Professor in Mathematics, University of Essex
Dedicated by the Author
to the Publisher
This book may be ordered from the printers, http://www.lulu.com/buy.
First published in 2000
by Torres Fremlin, 25 Ireton Road, Colchester CO3 3AT, England
(cid:13)c D.H.Fremlin 2000
The right of D.H.Fremlin to be identified as author of this work has been asserted in accordance with the Copyright,
Designs and Patents Act 1988. This work is issued under the terms of the Design Science License as published
in http://www.gnu.org/licenses/dsl.html. For the source files see http://www.essex.ac.uk/maths/people/
fremlin/mt1.2011/index.htm.
Library of Congress classification QA312.F72
AMS 2010 classification 28-01
ISBN 978-0-9538129-8-1
Typeset by AMS-TEX
Printed by Lulu.com
5
Contents
General Introduction 6
Introduction to Volume 1 7
Chapter 11: Measure Spaces
Introduction 9
111 σ-algebras 10
Definitionofσ-algebra;countablesets;σ-algebrageneratedbyafamilyofsets;Borelσ-algebras.
112 Measure spaces 14
Definition of measure space; the use of ∞; elementary properties; negligible sets; point-supported measures; image
measures.
113 Outer measures and Carath´eodory’s construction 19
Outermeasures;Carath´eodory’sconstructionofameasurefromanoutermeasure.
114 Lebesgue measure on R 23
Half-openintervals;Lebesgueoutermeasure;Lebesguemeasure;Borelsetsaremeasurable.
115 Lebesgue measure on Rr 28
Half-openintervals;Lebesgueoutermeasure;Lebesguemeasure;Borelsetsaremeasurable.
Chapter 12: Integration
Introduction 35
121 Measurable functions 35
Subspace σ-algebras; measurable real-valued functions; partially defined functions; Borel measurable functions; op-
erationsonmeasurablefunctions;generatingBorelsetsfromhalf-spaces.
122 Definition of the integral 43
Simplefunctions;non-negativeintegrablefunctions;integrablereal-valuedfunctions;virtuallymeasurablefunctions;
linearityoftheintegral.
123 The convergence theorems 52
B.Levi’stheorem;Fatou’slemma;Lebesgue’sDominatedConvergenceTheorem;differentiatingthroughanintegral.
Chapter 13: Complements
Introduction 56
131 Measurable subspaces 56
Subspacemeasuresonmeasurablesubsets;integrationovermeasurablesubsets.
132 Outer measures from measures 58
Theoutermeasureassociatedwithameasure;Lebesgueoutermeasureagain;measurableenvelopes.
133 Wider concepts of integration 61
∞asavalueofanintegral;complex-valuedfunctions;upperandlowerintegrals.
134 More on Lebesgue measure 68
Translation-invariance;non-measurablesets;innerandouterregularity;theCantorsetandfunction;*theRiemann
integral.
135 The extended real line 79
The algebra of ±∞; Borel sets and convergent sequences in [−∞,∞]; measurable and integrable [−∞,∞]-valued
functions.
*136 The Monotone Class Theorem 84
Theσ-algebrageneratedbyafamilyI;algebrasofsets.
Appendix to Volume 1
Introduction 89
1A1 Set theory 89
Notation;countableanduncountablesets.
1A2 Open and closed sets in Rr 92
Definitions;basicpropertiesofopenandclosedsets;Cauchy’sinequality;openballs.
1A3 Lim sups and lim infs 94
limsupn→∞an,liminfn→∞an in[−∞,∞].
6
Concordance 97
References for Volume 1 97
Index to Volume 1
Principal topics and results 98
General index 99
General Introduction
In this treatise I aim to give a comprehensive description of modern abstract measure theory, with some indication
ofitsprincipalapplications. Thefirsttwovolumesaresetatanintroductorylevel; theyareintendedforstudentswith
a solid grounding in the concepts of real analysis, but possibly with rather limited detailed knowledge. The emphasis
throughout is on the mathematical ideas involved, which in this subject are mostly to be found in the details of the
proofs.
My intention is that the book should be usable both as a first introduction to the subject and as a reference work.
For the sake of the first aim, I try to limit the ideas of the early volumes to those which are really essential to the
development of the basic theorems. For the sake of the second aim, I try to express these ideas in their full natural
generality,andinparticularItakecaretoavoidsuggestinganyunnecessaryrestrictionsintheirapplicability. Ofcourse
these principles are to to some extent contradictory. Nevertheless, I find that most of the time they are very nearly
reconcilable, provided that I indulge in a certain degree of repetition. For instance, right at the beginning, the puzzle
arises: should one develop Lebesgue measure first on the real line, and then in spaces of higher dimension, or should
one go straight to the multidimensional case? I believe that there is no single correct answer to this question. Most
studentswillfindtheone-dimensionalcaseeasier,anditthereforeseemsmoreappropriateforafirstintroduction,since
even in that case the technical problems can be daunting. But certainly every student of measure theory must at a
fairly early stage come to terms with Lebesgue area and volume as well as length; and with the correct formulations,
the multidimensional case differs from the one-dimensional case only in a definition and a (substantial) lemma. So
what I have done is to write them both out (in §§114-115), so that you can pass over the higher dimensions at first
reading (by omitting §115) and at the same time have a complete and uncluttered argument for them (if you omit
section §114). In the same spirit, I have been uninhibited, when setting out exercises, by the fact that many of the
results I invite students to look for will appear in later chapters; I believe that throughout mathematics one has a
better chance of understanding a theorem if one has previously attempted something similar alone.
The plan of the work is as follows:
Volume 1: The Irreducible Minimum
Volume 2: Broad Foundations
Volume 3: Measure Algebras
Volume 4: Topological Measure Spaces
Volume 5: Set-theoretic Measure Theory.
Volume1isintendedforthosewithnopriorknowledgeofmeasuretheory,butcompetentintheelementarytechniques
of real analysis. I hope that it will be found useful by undergraduates meeting Lebesgue measure for the first time.
Volume 2 aims to lay out some of the fundamental results of pure measure theory (the Radon-Nikody´m theorem,
Fubini’s theorem), but also gives short introductions to some of the most important applications of measure theory
(probability theory, Fourier analysis). While I should like to believe that most of it is written at a level accessible
to anyone who has mastered the contents of Volume 1, I should not myself have the courage to try to cover it in an
undergraduate course, though I would certainly attempt to include some parts of it. Volumes 3 and 4 are set at a
ratherhigherlevel,suitabletopostgraduatecourses; whileVolume5willassumeawide-rangingcompetenceoverlarge
parts of analysis and set theory.
There is a disclaimer which I ought to make in a place where you might see it in time to avoid paying for this book.
I make no attempt to describe the history of the subject. This is not because I think the history uninteresting or
unimportant; rather, it is because I have no confidence of saying anything which would not be seriously misleading.
IndeedIhaveverylittleconfidenceinanythingIhaveeverreadconcerningthehistoryofideas. SowhileIamhappyto
honour the names of Lebesgue and Kolmogorov and Maharam in more or less appropriate places, and I try to include
in the bibliographies the works which I have myself consulted, I leave any consideration of the details to those bolder
and better qualified than myself.
Introduction to Volume 1 7
For the time being, at least, printing will be in short runs. I hope that readers will be energetic in commenting on
errors and omissions, since it should be possible to correct these relatively promptly. An inevitable consequence of this
is that paragraph references may go out of date rather quickly. I shall be most flattered if anyone chooses to rely on
this book as a source for basic material; and I am willing to attempt to maintain a concordance to such references,
indicating where migratory results have come to rest for the moment, if authors will supply me with copies of papers
which use them. In the concordance to the present volume you will find notes on the items which have been referred
to in other published volumes of this work.
I mention some minor points concerning the layout of the material. Most sections conclude with lists of ‘basic
exercises’ and ‘further exercises’, which I hope will be generally instructive and occasionally entertaining. How many
of these you should attempt must be for you and your teacher, if any, to decide, as no two students will have quite
the same needs. I mark with a >>> those which seem to me to be particularly important. But while you may not need
to write out solutions to all the ‘basic exercises’, if you are in any doubt as to your capacity to do so you should take
this as a warning to slow down a bit. The ‘further exercises’ are unbounded in difficulty, and are unified only by a
presumption that each has at least one solution based on ideas already introduced.
Theimpulsetowritethistreatiseisinlargepartadesiretopresentaunifiedaccountofthesubject. Cross-references
arecorrespondinglyabundantandwide-ranging. Inordertobeabletoreferfreelyacrossthewholetext, Ihavechosen
a reference system which gives the same code name to a paragraph wherever it is being called from. Thus 132E is the
fifth paragraph in the second section of Chapter 13, which is itself the third chapter of this volume, and is referred to
by that name throughout. Let me emphasize that cross-references are supposed to help the reader, not distract him.
Do not take the interpolation ‘(121A)’ as an instruction, or even a recommendation, to turn back to §121. If you are
happy with an argument as it stands, independently of the reference, then carry on. If, however, I seem to have made
rather a large jump, or the notation has suddenly become opaque, local cross-references may help you to fill in the
gaps.
Eachvolumewillhaveanappendixof‘usefulfacts’,inwhichIsetoutmaterialwhichiscalledonsomewhereinthat
volume, and which I do not feel I can take for granted. Typically the arrangement of material in these appendices is
directed very narrowly at the particular applications I have in mind, and is unlikely to be a satisfactory substitute for
conventional treatments of the topics touched on. Moreover, the ideas may well be needed only on rare and isolated
occasions. So as a rule I recommend you to ignore the appendices until you have some direct reason to suppose that a
fragment may be useful to you.
During the extended gestation of this project I have been helped by many people, and I hope that my friends and
colleagues will be pleased when they recognise their ideas scattered through the pages below. But I am especially
grateful to those who have taken the trouble to read through earlier drafts and comment on obscurities and errors.
In particular, I should like to single out F.Nazarov and P.Wallace Thompson, whose thorough reading of the present
volume corrected many faults.
Introduction to Volume 1
In this introductory volume I set out, at a level which I hope will be suitable for students with no prior knowledge
of the Lebesgue (or even Riemann) integral and with only a basic (but thorough) preparation in the techniques of
(cid:15)-δ analysis, the theory of measure and integration up to the convergence theorems (§123). I add a third chapter
(Chapter 13) of miscellaneous additional results, mostly chosen as being relatively elementary material necessary for
topics treated in Volume 2 which does not have a natural place there.
The title of this volume is a little more emphatic than I should care to try to justify au pied de la lettre. I would
certainly characterise the construction of Lebesgue measure on R (§114), the definition of the integral on an abstract
measure space (§122) and the convergence theorems (§123) as indispensable. But a teacher who wishes to press on
to further topics will find that much of Chapter 13 can be set aside for a while. I say ‘teacher’ rather than ‘student’
here, because if you are studying on your own I think you should aim to go slower than the text requires rather than
faster; in my view, these ideas are genuinely difficult, and I think you should take the time to get as much practice at
relatively elementary levels as you can.
Perhaps this is a suitable moment at which to set down some general thoughts on the teaching of measure theory. I
havebeenteachinganalysisforoverthirtyyearsnow,andoneofthefewconstantsoverthatperiodhasbeenthefeeling,
almostuniversalamongteachersofanalysis,thatwearenotservingmostofourstudentswell. Wehaveallencountered
students who are not stupid – who are indeed quite good at mathematics – but who seem to have a disproportionate
difficulty with rigorous analysis. They are so exhausted and demoralised by the technical problems that they cannot
makesenseoruseevenoftheknowledgetheyachieve. Thenaturalreactiontothisistotrytomakecoursesshorterand
easier. But I think that this makes it even more likely that at the end of the semester your students will be stranded
in thorn-bushes half way up the mountain. Specifically, with Lebesgue measure, you are in danger of spending twenty
8 Introduction to Volume 1
hours teaching them how to integrate the characteristic function of the rationals. This is not what the subject is for.
Lebesgue’s own presentations of the subject (Lebesgue 1904, Lebesgue 1918) emphasize the convergence theorems
and the Fundamental Theorem of Calculus. I have put the former in Volume 1 and the latter in Volume 2, but it does
seem to me that unless your students themselves want to know when one can expect to be able to interchange a limit
and an integral, or which functions are indefinite integrals, or what the completions of C([0,1]) under the norms (cid:107)(cid:107) ,
1
(cid:107)(cid:107) look like, then it is going to be very difficult for them to make anything of this material; and if you really cannot
2
reach the point of explaining at least a couple of these matters in terms which they can appreciate, then it may not be
worth starting. I would myself choose rather to omit a good many proofs than to come to the theorems for which the
subject was created so late in the course that two thirds of my class have already given up before they are covered.
OfcourseIandothershavefollowedthatroadtoo,withnobetterresults(thoughusuallywithhappierstudents)than
we obtain by dotting every i and crossing every t in the proofs. Nearly every time I am consulted by a non-specialist
who wants to be told a theorem which will solve his problem, I am reminded that pure mathematics in general, and
analysis in particular, does not lie in the theorems but in the proofs. In so far as I have been successful in answering
such questions, it has usually been by making a trifling adjustment to a standard argument to produce a non-standard
theorem. The ideas are in the details. You have not understood Carath´eodory’s construction (§113) until you can, at
the very least, reliably reproduce the argument which shows that it works. In the end, there is no alternative to going
overeverystepoftheground, andwhileIhaveoccasionallybeenruthlessincuttingouttopicswhichseemtometobe
marginal, I have tried to make sure – at the expense, frequently, of pedantry – that every necessary idea is signalled.
Faced,therefore,withanyparticularclass,Ibelievethatateachermustcompromisebetweenscopeandcompleteness.
Exactly which compromises are most appropriate will depend on factors which it would be a waste of time for me to
guess at. This volume is supposed to be a possible text on which to base a course; but I hope that no lecturer will
set her class to read it at so many pages a week. My primary aim is to provide a concise and coherent basis on which
to erect the structure of the later volumes. This involves me in pursuing, at more than one point, approaches which
take slightly more difficult paths for the sake of developing a more refined technique. (Perhaps the most salient of
these is my insistence that an integrable function need not be defined everywhere on the underlying measure space;
see §§121-122.) It is the responsibility of the individual teacher to decide for herself whether such refinements are
appropriate to the needs of her students, and, if not, to show them what translations are needed.
The above paragraphs are directed at teachers who are, supposedly, competent in the subject – certainly past the
level treated in this volume – and who have access to some of the many excellent books already available, so that if
they take the trouble to think out their aims, they should be able to choose which elements of my presentation are
suitable. ButImustalsoconsiderthepositionofastudentwhoissettingouttolearnthismaterialonhisown. Itrust
that you have understood from what I have already written that you should not be afraid to look ahead. You could,
indeed, do worse than go to Volume 2, and take one of the wonderful theorems there – the Fundamental Theorem of
Calculus (§222), for instance, or, if you are very ambitious, the strong law of large numbers (§273) – and use the index
and the cross-references to try to extract a proof from first principles. If you are successful you will have every right
to congratulate yourself. In the periods in which success seems elusive, however, you should be working systematically
throughthe‘basicexercises’inthesectionswhichseemtoberelevant; andifallelsefails, startagainatthebeginning.
Mathematics is a difficult subject, that is why it is worth doing, and almost every section here contains some essential
idea which you could not expect to find alone.
Note on second and third printings
For the second printing of this volume I made a few corrections, with a handful of new exercises. For the third
printing I have done the same; in addition, I have given an elementary extra result and formal definitions of some
almost standard terms. I have also allowed myself, in a couple of cases, to rearrange a set of exercises into what now
seems to me a more natural order.
Note on second edition, 2011
For the new (‘Lulu’) edition of this volume, I have eliminated a number of further errors; no doubt many remain.
There are some further exercises, and a little more material on upper and lower integrals (§133).
Chap. 11 intro. Introduction 9
Chapter 11
Measure spaces
In this chapter I set out the fundamental concept of ‘measure space’, that is, a set in which some (not, as a rule,
all) subsets may be assigned a ‘measure’, which you may wish to interpret as area, or mass, or volume, or thermal
capacity, or indeed almost anything which you would expect to be additive – I mean, that the measure of the union
of two disjoint sets should be the sum of their measures. The actual definition (in 112A) is not obvious, and depends
essentially on certain technical features which make a preparatory section (§111) advisable. Furthermore, even with
the definition well in hand, the original and most important examples of measures, Lebesgue measure on Euclidean
space, remain elusive. I therefore devote a section (§113) to a method of constructing measures, before turning to the
details of the arguments needed for Lebesgue measure in §§114-115. Thus the structure of the chapter is three sections
of general theory followed by two (which are closely similar) on particular examples. I should say that the general
theory is essentially easier; but it does rely on facility with certain manipulations of families of sets which may be new
to you.
At some point I ought to comment on my arrangement of the material, and it may be helpful if I do so before you
start work on this chapter. One of the many fundamental questions which any author on the subject must decide, is
whethertobeginwith‘general’measuretheoryorwith‘Lebesgue’measureandintegration. ThepointisthatLebesgue
measure is rather more than just the most important example of a measure space. It is so close to the heart of the
subject that the great majority of the ideas of the elementary theory can be fully realised in theorems about Lebesgue
measure. Looking ahead to Volume 2, I find that, with the exception of Chapter 21 – which is specifically devoted
to extending your ideas of what measure spaces can be – only Chapter 27 and the second half of Chapter 25 really
need the general theory to make sense, while Chapters 22, 26 and 28 are specifically about Lebesgue measure. Volume
3 is another matter, but even there more than half the mathematical content can be expressed in terms of Lebesgue
measure. Ifyoutaketheview,asIcertainlydowhenitsuitsmyargument,thatthebusinessofpuremathematicsisto
express and extend the logical capacity of the human mind, and that the actual theorems we work through are merely
vehiclesfortheideas,thenyoucancorrectlypointoutthatallthereallyimportantthingsinthepresentvolumecanbe
donewithoutgoingtothetroubleofformulatingageneraltheoryofabstractmeasurespaces; andthatbystudyingthe
relativelyconcreteexampleofLebesguemeasureonr-dimensionalEuclideanspaceyoucanavoidavarietyofirrelevant
distractions.
If you are quite sure, as a teacher, that none of your pupils will wish to go beyond the elementary theory, there
is something to be said for this view. I believe, however, that it becomes untenable if you wish to prepare any of
your students for more advanced ideas. The difficulty is that, with the best will in the world, anyone who has worked
through the full theory of Lebesgue measure, and then comes to the theory of abstract measure spaces, is likely to go
through it too fast, and at the end find himself uncertain about just which ninety per cent of the facts he knows are
generally applicable. I believe it is safer to keep the special properties of Lebesgue measure clearly labelled as such
from the beginning.
Itisofcoursethebesettingsinofmathematicsteachersatthislevel,toteachaclassoftwentyinamannerappropriate
to perhaps two of them. But in the present case my own judgement is that very few students who are ready for the
course at all will have any difficulty with the extra level of abstraction involved in ‘Let (X,Σ,µ) be a measure space,
...’. I do assume knowledge of elementary linear algebra, and the grammar, at least, of arbitrary measure spaces is
no worse than the grammar of arbitrary linear spaces. Moreover, the Lebesgue theory already involves statements of
theform‘ifE isaLebesguemeasurableset, ...’, and inmyexperiencestudentswhocan cope withquantificationover
subsets of the reals are not deterred by quantification over sets of sets (which anyway is necessary for any elementary
description of the σ-algebra of Borel sets). So I believe that here, at least, the extra generality of the ‘professional’
approach is not an obstacle to the amateur.
I have written all this here, rather than later in the chapter, because I do wish to give you the choice. And if your
choice is to learn the Lebesgue theory first, and leave the general theory to later, this is how to do it. You should read
paragraphs 114A-114C
114D, with 113A-113B and 112Ba, 112Bc
114E, with 113C-113D, 111A, 112A, 112Bb
114F
114G, with 111G and 111C-111F,
and then continue with Chapter 12. At some point, of course, you should look at the exercises for §§112-113; but, as
in Chapters 12-13, you will do so by translating ‘Let (X,Σ,µ) be a measure space’ into ‘Let µ be Lebesgue measure
on R, and Σ the σ-algebra of Lebesgue measurable sets’. Similarly, when you look at 111X-111Y, you will take Σ to
be either the σ-algebra of Lebesgue measurable sets or the σ-algebra of Borel subsets of R.
10 Measure spaces §111 intro.
111 σ-algebras
In the introduction to this chapter I remarked that a measure space is ‘a set in which some (not, as a rule, all)
subsets may be assigned a measure’. All ordinary concepts of ‘length’ or ‘area’ or ‘volume’ apply only to reasonably
regularsets. Modernmeasuretheoryisremarkablypowerfulinthatanextraordinaryvarietyofsetsareregularenough
to be measured; but we must still expect some limitation, and when studying any measure a proper understanding of
theclassofsetswhichitmeasureswillbecentraltoourwork. Thebasicdefinitionhereisthatof‘σ-algebraofsets’;all
measures in the standard theory are defined on such collections. I therefore begin with a statement of the definition,
and a brief discussion of the properties, of these classes.
111A Definition Let X be a set. A σ-algebra of subsets of X (sometimes called a σ-field) is a family Σ of
subsets of X such that
(i) ∅∈Σ;
(ii) for every E ∈Σ, its complement X\E in X belongs to Σ;
(cid:83)
(iii) for every sequence (cid:104)En(cid:105)n∈N in Σ, its union n∈NEn belongs to Σ.
111B Remarks (a) Almost any new subject in pure mathematics is likely to begin with definitions. At this point
there is no substitute for rote learning. These definitions encapsulate years, sometimes centuries, of thought by many
people; you cannot expect that they will always correspond to familiar ideas.
(b)Nevertheless,youshouldalwaysseekimmediatelytofindwaysofmakingnewdefinitionsmoreconcretebyfinding
examples within your previous mathematical experience. In the case of ‘σ-algebra’, the really important examples, to
be described below, are going to be essentially new – supposing, that is, that you need to read this chapter at all.
However, two examples should be immediately accessible to you, and you should bear these in mind henceforth:
(i) for any X, Σ={∅,X} is a σ-algebra of subsets of X;
(ii) for any X, PX, the set of all subsets of X, is a σ-algebra of subsets of X.
Theseareofcoursethesmallestandlargestσ-algebrasofsubsetsofX, andwhileweshallspendlittletimewiththem,
both are in fact significant.
*(c) The phrase measurable space is often used to mean a pair (X,Σ), where X is a set and Σ is a σ-algebra of
subsets of X; but I myself prefer to avoid this terminology, unless greatly pressed for time, as in fact many of the most
interesting examples of such objects have no useful measures associated with them.
111C Infinite unions and intersections If you have not seen infinite unions before, it is worth pausing over the
(cid:83)
formula E . This is the set of points belonging to one or more of the sets E ; we may write it as
n∈N n n
(cid:91)
E ={x:∃ n∈N, x∈E }
n n
n∈N
=E ∪E ∪E ∪... .
0 1 2
(I write N for the set of natural numbers {0,1,2,3,...}.) In the same way,
(cid:92)
E ={x:x∈E ∀n∈N}
n n
n∈N
=E ∩E ∩E ∩... .
0 1 2
It is characteristic of the elementary theory of measure spaces that it demands greater facility with the set-operations
∪, ∩, \ (‘set difference’: E \F = {x : x ∈ E, x ∈/ F}), (cid:52) (‘symmetric difference’: E(cid:52)F = (E \F)∪(F \E) =
(E∪F)\(E∩F))thanyouhaveprobablyneededbefore,withtheaddedcomplicationofinfiniteunionsandintersections.
I strongly advise spending at least a little time with Exercise 111Xa at some point.
111D Elementary properties of σ-algebras If Σ is a σ-algebra of subsets of X, then it has the following
properties.
(a) E∪F ∈Σ for all E, F ∈Σ. PPP For if E, F ∈Σ, set E0 =E, En =F for n≥1; then (cid:104)En(cid:105)n∈N is a sequence in
(cid:83)
Σ and E∪F = E ∈Σ. QQQ
n∈N n
(b)E∩F ∈ΣforallE, F ∈Σ. PPPBy(ii)ofthedefinitionin111A,X\E andX\F ∈Σ; by(a)ofthisparagraph,
(X\E)∪(X\F)∈Σ; by 111A(ii) again, X\((X\E)∪(X\F))∈Σ; but this is just E∩F. QQQ